I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here:
The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example of the application of this theorem:
I am unable to follow the logic in this example. My understanding of the theorem is that if two closed 2-forms agree on some submanifold $N$, then they agree up to some diffeomorphism in some neighborhood around it. My logic in trying to follow the example goes something like this:
- We start by picking a choice of coordinates around $m$ and defining a map from points around $0$ in $T_m M$ with points around $m$ in $M$.
- We obtain a symplectic form $\omega_0$ by using the basis for $T_m M$ induced by the choice of coordinates around $m$ (i.e, if our choice of coordinates around $m$ is $x^i$, $i=1\ldots 2\ell$, then $\omega_0 = \sum_{i=1}^\ell dx^i \wedge dx^{i+\ell} $)
- If $G$ fixes the point $m$, then $G$ has some simple linear action on $T_m M$, since you can think of $T_m M$ as equivalence classes of curves passing through $m$, and the linear action on $T_m M$ is obtained by just seeing how $G$ warps these curves.
Past this, I'm stumped. I've tried to be explicit about the assumptions I'm making because my issue probably lies in one of those assumptions. Specifically, I'm not sure how the theorem relating forms to each other can imply something about the linear action of $G$ on the coordinates themselves. I'm assuming this has something to do with the fact that we have a map from a neighborhood of $0$ in $T_m M$ to a neighborhood of $m$ in $M$, and we know that $G$ acts linearly on the former; however, I don't see how it having a linear action on the latter is implied by the theorem. And what is this special coordinate system they talk about "with respect to which $\omega_0$ is the standard antisymmetric form on a symplectic vector space and the action of $G$ is linear"? Is this coordinate system just the one that we used to define $\omega_0$? Additionally, it seems like the choice of $\omega_0$ is arbitrary, so I don't know why we would expect $(\omega_0)|_N = (\omega_1)|_N$.
Consider $m$ as a submanifold of $M$, then $\omega_0|_m=\omega_1|_m=0$. Thus by the theorem, there is a neighbourhood and such a $\phi$ brings $\omega_1$ to $\omega_0$ on the neighbourhood.
The linearity of $G$ is a consequence of $m$ being a fixed point of $G$. For an element $g\in G$, the map on $T_mM$ is just the differential of $g$.
PS. I don't see any equivariant about $G$ in this case.